gyarfas_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Gyárfás conjecture (1975): For each t, every graph with no induced path on t vertices has chromatic number bounded above by a function of its clique number. Proved for t ≤ 4 and for chordal graphs.

Why this node was retired

The posted statement is

import Mathlib

-- Gyárfás conjecture: P_t-free graphs have χ bounded by a function of ω (clique number)
theorem gyarfas_conjecture :
    ∀ t : ℕ, ∃ f : ℕ → ℕ,
    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj]
      (ω : ℕ) (hω : ∀ S : Finset (Fin n), G.IsClique (S : Set (Fin n)) → S.card ≤ ω),
      (¬∃ path : Fin t → Fin n, Function.Injective path ∧
        ∀ i j : Fin t, i.val + 1 = j.val → G.Adj (path i) (path j)) →
      G.chromaticNumber ≤ f ω := by
  sorry

The forbidden path is ordinary, not induced.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Require that the injective path map has adjacency iff the two indices are consecutive; then quantify the color bound in terms of clique number.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.